Curvature formula, part 1

So, in the last video, I talked about curvature and the radius of curvature. I described it purely geometrically, where I'm saying you imagine driving along a certain road. Your steering wheel locks, and you're wondering what the radius of the circle that you draw with your car, you know, through whatever surrounding fields there are on the road as a result.

The special symbol that we have for this idea of curvature is a little kappa, and that's going to be one divided by the radius. The reason we do that is basically you want a large kappa, a large curvature, to correspond with a sharp turn. So, sharp turn, small radius, large curvature.

But the question is, how do we describe this in a more and more mathematical way? I'm going to go ahead and clear up, get rid of the circle itself and all of that radius, and just be looking at the curve itself in its own right.

The way you typically describe a curve like this is parametrically. So, you'll have some kind of vector-valued function s that takes in a single parameter t, and then it's going to output the x and y coordinates each as functions of t.

In this specific case, I'll just tell you the curve that I drew happens to be parameterized by 1 minus the sine of t as the x component function—actually, no, it's t minus sine of t—and the bottom part is 1 minus cosine of t. That's the curve that I drew.

The way that you're thinking about this is for each value t, you get a certain vector that puts you at a point on the curve. As t changes, the vector you get changes, but the tip of that vector kind of traces out the curve as a whole, right? You know, maybe you could imagine just the vector drawing the curve as t varies.

The thought behind making curvature mathematical here—I'll kind of clear up some room for myself—is that you take the tangent vectors here. You might imagine like a unit tangent vector at every given point, and you're wondering how quickly those guys change direction.

With the little schematic that I have drawn here, I might call this guy—I'm just going to call this guy t1 for like the first tangent vector, t2, t3—and I haven't, you know, specified where they start or anything. I just want to give a feel for you've got various different tangent vectors.

I'm just going to say that all of them, each one of those t sub somethings has a unit length. The idea of curvature is to look at how quickly that unit tangent vector changes directions. So, you might imagine just a completely different space.

Rather than rooting each vector on the curve, let’s see what it would look like if you just kind of write each vector in its own right, off in some other spot. So, this guy here would be t1, and then t2 points a little bit— a little bit down, and then t3 points kind of much, much more down.

So, all of these—this would be t1, that guy is t2—and these are the same vectors; I'm just kind of drawing them all rooted at the same spot so it's a little easier to see how they turn. You want to say, okay, how much do you change as you go from t1 to t2? Is that a large angle change? And as you go from t2 to t3, is that a large change as well?

You can kind of see how if you have a curve, and let’s say it's if you have a curve that curves quite a bit—you know, it's doing something like that—then the unit vector, the unit tangent vector at this point changes quite rapidly over a short distance to be something almost 90 degrees different.

Whereas if you take the unit vector here and see how much it changes as you go from this point over to this point, it doesn't really change that much. So, the thought behind curvature is we’re going to take the rate of change of that unit tangent vector.

I’m going to let capital T be a function that tells you whatever the unit tangent vector at each point is. And I'm not going to take the rate of change in terms of, you know, the parameter little t which is what we use to parameterize the curve because it shouldn't matter how you parameterize the curve.

Maybe you're driving along it really quickly or really slowly. Instead, what you want to take is the rate of change with respect to arc length. Arc length, and I'm using the variable s here to denote arc length.

What I mean by that is if you take just a tiny little step here, the distance of that step, the actual distance in the x, y plane, you'd consider to be the arc length. If you imagine it being really, really small, you're considering that a ds—a tiny change in the arc length.

So, this is the quantity that we care about: how much does that unit tangent vector change with respect to a tiny change in the arc length? You know, as we travel along, let’s say it was a distance of like 0.5, right? You want to know: did this unit vector change a lot or a little bit?

But I should add something here: it's this tiny change in the vector that would tell you, you know, what the vector connecting their two tips is. So, this would be a vector-valued quantity, and curvature itself should just be a number.

So, what we really care about is the size of this guy, right? So, what we really care about is the size of this, which is a vector-valued quantity, and that'll be an indication of just how much the curve curves.

But if, on a sharper turned curve, you go over that same distance, and then suddenly the change in the tangent vectors goes by quite a bit, that would be telling you it's a high curvature.

In the next video, I'm going to talk through what that looks like—how you think about this tangent vector function, this unit tangent vector function, and what it looks like to take the derivative of that with respect to arc length.

It can get a little convoluted in terms of the symbols involved, and the constant picture you should have in the back of your mind is that circle—that circle that's hugging the curve very closely at a certain point.

This means of taking the magnitude of the rate of change of the unit tangent vector with respect to arc length is all just trying to capture that idea.